Applied Energy 138 (2015) 496–504
Contents lists available at ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy
An inverse method for calculation of thermal inertia and heat gain
in air conditioning and refrigeration systems
M.A. Fayazbakhsh ⇑, F. Bagheri, M. Bahrami
Laboratory for Alternative Energy Conversion (LAEC), School of Mechatronic Systems Engineering, Simon Fraser University, 250-13450 102 Avenue, Surrey, BC V3T 0A3, Canada
h i g h l i g h t s
An inverse method is proposed to calculate thermal inertia in HVAC-R systems.
Real-time thermal loads are estimated using the proposed intelligent algorithm.
Calculation algorithm is validated with on-site measurements.
Freezer duty cycle data are extracted only based on temperature measurements.
a r t i c l e
i n f o
Article history:
Received 9 February 2014
Received in revised form 25 October 2014
Accepted 1 November 2014
Available online 16 November 2014
Keywords:
HVAC-R
Thermal inertia
Heat gain
Inverse method
a b s t r a c t
A new inverse method is proposed for estimation of thermal inertia and heat gain in air conditioning and
refrigeration systems using on-site temperature measurements. The method is applied on a walk-in freezer room of a restaurant in Surrey, British Columbia, Canada during one week of its regular operation. The
thermal inertia and instantaneous heat gain are calculated and the results are validated using actual
information of the materials inside the freezer room. The proposed method can be implemented in intelligent control systems designed for new and existing HVAC-R systems to improve their overall energy
efficiency and reduce their environmental impacts.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Heating, Ventilation, Air Conditioning, and Refrigeration
(HVAC-R) contribute to a tremendous portion of the global energy
consumption in a wide array of residential, industrial, and commercial applications worldwide. HVAC-R consumes half of the
energy use in buildings and 20% of the total national energy use
in European and American countries [1]. Predictions indicate a further increase of 50% from the current figure during the next
15 years in the European Union countries [1]. Furthermore, Air
Conditioning (AC) is the second most energy consuming unit in
vehicles [2]. In an electric vehicle, AC power consumption can be
as high as 12% of the total vehicle power during regular commuting
[3]. As such, efficient design of new HVAC-R systems and devising
intelligent control methods for existing systems can lead to significant reduction of total energy consumption and greenhouse gas
emissions in large scales.
⇑ Corresponding author. Tel.: +1 778 782 8538.
E-mail addresses: mfayazba@sfu.ca (M.A. Fayazbakhsh), mbahrami@sfu.ca
(M. Bahrami).
http://dx.doi.org/10.1016/j.apenergy.2014.11.004
0306-2619/Ó 2014 Elsevier Ltd. All rights reserved.
Proper design and intelligent operation of an HVAC-R system
requires: (i) accurate prediction of thermal loads, and (ii) appropriate design and selection of the cooling unit. Accurate cooling load
calculations are a prerequisite to proper sizing, selection, and control of any HVAC-R system. Modern air conditioning systems are
equipped with feedback controllers that allow them to manipulate
the operation of the HVAC unit in order to efficiently sustain thermal comfort. Although parameters such as room temperature,
humidity, and occupancy level are used as control variables, it is
advantageous to extend the first design step, i.e., the load estimation to the operation stage of the unit. Real-time estimation and
prediction of the upcoming thermal loads alongside raw measurements can be beneficial for energy-efficient control of HVAC systems, especially in Mobile Air Conditioning (MAC) applications
that experience highly dynamic load variations.
In many applications, controllers based on on–off action or
modulating control are successfully used that utilize the room
temperature as the controlled variable [4]. Nevertheless, it is inherently impossible to predict upcoming thermal conditions by using
conventional controllers that only rely on the current room status.
It is shown that intelligent control of the HVAC operation based on
M.A. Fayazbakhsh et al. / Applied Energy 138 (2015) 496–504
prediction of the load can help maintain air quality while minimizing energy consumption [5,6]. By predicting thermal loads, controllers are enabled to not only provide thermal comfort in the current
condition, but also adjust the system operation to cope with the
upcoming conditions in an efficient manner. Thus, improvement
of load calculation methods and the ability to estimate and predict
the loads in real-time can improve the feedback information for
the control of the HVAC system, which, in turn, result in significant
reduction of total energy consumption and greenhouse gas emissions. Furthermore, in applications where the room contents may
vary over time, an algorithm that can estimate the thermal inertia
in an unsupervised manner can aid the HVAC-R system to adapt to
the new conditions. Given the automatic adaptability of the HVACR system, it can be further controlled to perform at different capacities in order to reduce the overall energy consumption.
The American Society of Heating, Refrigerating, and Air Conditioning Engineers (ASHRAE) established an extensive methodology
for calculation of heating and cooling loads. The heat balance
method [7,8] is an example of such methods. It is a straight-forward and rigorous method that involves calculating a surface-bysurface heat balance of the surrounding walls of a room through
consideration of all conductive, convective, and radiative heat
transfer mechanisms. The method has been extensively used in
residential and non-residential applications [5,6]. Mobile Air Conditioning (MAC), especially in electric vehicles, is also a crucial
application where potential HVAC energy savings are possible. Less
energy consumption by mobile HVAC systems directly results in
higher mileage and better overall efficiency on the road [9]. Zheng
et al. [10] devised a simple method to calculate thermal loads in
vehicles. In their approach the different load categories such as
radiation and ambient loads were considered. A case study was
performed and the results were validated by experiments.
Although their methodology has shown good agreement with
experimental results, much details of the vehicle cabin and service
conditions is needed to perform the necessary modelling calculations. Arici et al. [11] developed a computer code for simulation
of the dynamic operation of a climate control system in a typical
vehicle. Khayyam et al. [12] collected a set of models to calculate
the different categories of thermal loads encountered in a vehicle.
After using appropriate models for calculation of each of the load
categories, they implemented control algorithms to improve the
overall efficiency of the mobile air conditioning system.
In many applications, thermal loads vary dynamically. Novel
approaches are being studied in the literature for real-time estimation of thermal loads. Several methodologies are introduced in the
literature for estimation of thermal loads in existing buildings.
Kashiwagi and Tobi [13] proposed a neural network algorithm
for prediction of thermal loads. Ben-Nakhi and Mahmoud [14] also
used general regression neural networks and concluded that a
properly-designed neural network is a powerful tool for optimizing
thermal energy storage in buildings based only on external temperature records. They claimed that their set of algorithms could
learn over time and improve the prediction ability. Li et al. [15]
presented four modeling techniques for hourly prediction of cooling loads. The methods included back propagation neural network
(BPNN), radial basis function neural network (RBFNN), general
regression neural network (GRNN), and support vector machine
(SVM). Other researchers have used fuzzy control algorithms to
propose load prediction methods. Among many, Sousa et al. [16]
developed a fuzzy controller to be incorporated as a predictor in
a nonlinear model-based predictive controller. Wang and Xu
[17,18] used a Genetic Algorithm (GA) for identification of parameters in a model for estimation of thermal performance. In their
approach, a thermal network of lumped thermal masses and
parameters was identified using operation data and GA estimators.
497
A disadvantage of most existing load calculation methods is
that they require much information about the air-conditioned
space to estimate the loads. For instance, the heat balance method
requires knowledge of material properties, thickness of walls, geographical location, fenestration data, weather information, occupancy, appliances, and other detailed information. Such an
approach rarely relies on feedback information from the existing
air-conditioned space. That type of methodology is a ‘‘forward’’
approach which makes the redesign/retrofit of existing HVAC-R
systems a laborious and time-consuming task.
Alongside forward approaches, ASHRAE also recognizes datadriven or ‘‘inverse’’ methods of load calculation [7]. The data-driven modelling methodology consists of gathering the performance
data of an existing system and analyzing them. Relatively few
parameters are required in an inverse approach compared to forward methodologies. In an inverse method, model parameters
may be deduced from the room data. Thus, the inverse model
can predict the ‘‘as-built’’ system performance more accurately
[7]. Inverse methods concentrate on the study of existing HVACR systems and allow the thermal performance of the system to
be inferred based on measured temperature data. This is particularly convenient for retrofitting existing systems. Major input data
to an inverse algorithm are the room temperature under regular
operation, as well as the performance/capacity of the HVAC-R system. Therefore, in an inverse method, the entire system, i.e., the
conditioned space plus the HVAC-R unit is merely seen as a black
box that is investigated for a period of time.
In this study, a new inverse methodology is proposed for identification of the duty cycles in a typical refrigeration system. The
lumped thermal inertia and heat gains are quantitatively calculated. The present approach is scalable and is irrespective to the
shape of the conditioned room. Therefore, it can be applied to
any HVAC-R application. The proposed analysis enables an accurate and real-time calculation of thermal loads. Therefore, it can
be used for intelligent control of the corresponding HVAC-R unit.
Many existing HVAC-R devices are equipped with constant-speed
compressors and fans. Nevertheless, Qureshi and Tassou [19]
reviewed the application of variable-speed capacity control in
refrigeration systems. They argued that in order to compensate
for half-load usage conditions, the option of variable-speed compressor consumed the least percentage of the full load power compared to other methods. Hence, implementation of the present
algorithm combined with variable-speed compressors can further
reduce the energy consumption and environmental impact of
HVAC-R systems.
In the following sections, the acquisition of experimental data is
first explained. Based on the collected data of the sample refrigeration system, the proposed model is discussed in a subsequent section. Results of the modelling approach are finally presented and
the accuracy of the results is validated using experimental information of the freezer room.
2. Experimental study
A freezer room in a restaurant in Surrey, British Columbia,
Canada was selected to collect data during its regular operation.
Fig. 1 shows a picture of the inside of the walk-in freezer room
as well as a 3D schematic of it. Dimensions of the room are shown
in Fig. 1 as well.
Temperature sensors were installed in different locations inside
the freezer room. Temperature data loggers (Track-It, Monarch
Instruments) were used for logging the room temperature over a
period of one week. The maximum error of the temperature data
loggers is ±1 °C for the range of 20 °C to 85 °C. Fig. 2 shows
the 7 locations where temperature data loggers were installed.
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230cm
270cm
250cm
Left
Front
(a)
(b)
Fig. 1. (a) A picture of the freezer room. (b) 3D schematic of the freezer room with dimensions.
Evaporator
RT5
RT1
Door lamp
RT3
RT2
RT7
RT4
Room Temperature, T (°C)
RT6
5
0
-5
-10
-15
-20
0
Door
Shelves
1
2
3
4
5
6
7
Time, t (day)
Fig. 2. Location of temperature data loggers in the freezer room. RT1: above door,
RT2: left shelf, RT3: back shelf, RT4: right shelf, RT5: ceiling, RT6: beside door lamp,
RT7: beside door hinge.
Fig. 3. Average temperature in the freezer room from 28 August 2013 at 00:01 am
until 3 September 2013 at 11:59 pm. Arrows highlight noticeable temperature
swings between low and high temperature set points, brackets show heavy duty
periods, and circles denote temperature spikes due to defrosting.
Temperature was measured in various locations to determine its
non-uniformity inside the room. It was observed that a temperature non-uniformity of less than 2 °C existed throughout the room.
Therefore, an average value was used in the analysis to represent
the lumped freezer room temperature.
Variations of the freezer room average temperature are shown
in Fig. 3. The data show one week of freezer regular operation from
28 August 2013 at 00:01 am until 3 September 2013 at 11:59 pm.
The freezer room was studied during 5 months from August 2013
to December 2013. Similar trends were observed throughout the
period of study, since the freezer is placed in an air-conditioned
indoor restaurant area and there is negligible direct influence from
the outdoor temperature on the room performance. Therefore, the
above-mentioned weeklong data is selected for the present
analysis.
The freezer room in this study, as well as most existing refrigeration units, is controlled based on low and high temperature set
points. Consequently, the system produces cooling until the room
temperature reaches the low temperature set point. Once the minimum temperature is reached, the thermostat shuts the refrigeration unit down. At this point, the decrease of the room
temperature stops and the temperature begins to increase as a
result of heat gains. Consequently, the temperature increases until
it reaches the high temperature set point. Once this maximum tolerable set point is reached, the thermostat triggers the refrigeration unit to switch back on. The low and high temperature set
points are settings of the room thermostat and they can also be
inferred from the temperature measurements, Fig. 3. By using the
measured temperature at a location other than the thermostat,
there can be a discrepancy between the actual thermostat set
points and the apparent set points observed from the measurements. This is due to the temperature non-uniformity within the
freezer because of which the set points may seem to vary on different days as well. To keep the generality of the approach for systems
with the least available information, the apparent set points
observed on the temperature data are used in this study.
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Investigating the temperature graph shown in Fig. 3, the following 3 patterns are noticeable:
1. Swinging regimes
Temperature oscillations or swings occur mainly due to the
starts/stops of the refrigeration unit. The period of these oscillations in the current application is approximately 20 min. Arrows
show regions of swinging regime in Fig. 3. Fig. 4 shows a zoomed
view of these oscillations. The apparent low and high temperature
set points are identified from Fig. 4 to be 15.1 °C and 13.8 °C,
respectively.
by irregular and random-looking variations in the freezer temperature. Since the temperature is mostly above the set points, it
means that the condensing unit has been constantly on during
these periods. Heavy duty periods depend on the usage pattern
and are demarcated using brackets in Fig. 3. It is observed in
Fig. 3 that the irregularities in temperature rise mostly occurred
during daytimes of the first 4 days of measurements. In contrary,
the next 3 days mostly showed swinging regimes and defrost
spikes. It can be concluded from Fig. 3 that door openings and loading/unloading of goods in the freezer room considerably contribute
to changes in the heat gain of the freezer room.
3. Present model
2. Temperature spikes
Rapid temperature increases are noticeable in some areas of
Fig. 3. They are due to the occurrence of defrosting in the freezer.
Defrosting is a process that melts the frost away from evaporator
coils and is unavoidable for most systems. The defrost system
can work either by heating the evaporator coils or turning off the
system [20]. An automatic defrost system exists in the studied
freezer room which kicks in every 6 h and heats up the evaporator
coils to melt the frost. The defrost events, even though a necessity,
impose a huge amount of heat load on the freezer room, as perceived by the sharp temperature increases in Fig. 3.
3. Heavy duty periods
Periods of heavy duty imposed on the refrigeration unit are also
visible in certain times of specific days. In Fig. 3 they are denoted
A lumped thermal model of the room is used to develop the
present inverse model. The cold room is assumed as a black box
of which little information is available. In contrary to the forward
approach, the objective is to determine the thermal characteristics
of the freezer room based on available experimental data. A
lumped or zero-dimensional thermal model can be described by
two characteristics of the freezer enclosure: (i) total thermal inertia and (ii) total instantaneous heat gain. The contributing components to the room heat gain can be categorized into the following:
1. Direct gain: from electrical equipment, light bulbs, human metabolic loads, etc.
2. Ambient gain: from convective and conductive heat transfer
across room walls.
3. Ventilation gain: from infiltration of air into the room.
4. Solar gain: from solar radiation into the room.
Room Temperature, T (°C)
-13
The summation of the above heat gains equals the overall heat
gain of the freezer room. The total instantaneous cooling load is not
necessarily equal to the total heat gain [7]. Nevertheless, the net
thermal energy transferred to the room contributes to the temperature variation rate of the cooled enclosure. Therefore, the cooling
load provided by the refrigeration system must satisfy the following heat balance equation:
−13.8°C
-14
dT
Q_ Cooling þ Q_ Gain ¼ M
dt
-15
−15.1°C
-16
180
200
220
240
260
280
300
Time, t (Min)
Fig. 4. Sample pattern of temperature swinging between low and high set points.
The data shows the room temperature on 2 September 2013 from 03:00 am until
05:00 am. Apparent low and high temperature set points are identified as 15.1 °C
and 13.8 °C, respectively.
Measuring the
room
temperature
over time
Smoothing the
temperature
variation
diagram
ð1Þ
where Q_ Gain is the total heat gain, Q_ Cooling is the instantaneous cooling load provided by the refrigeration unit, M is the overall thermal
inertia of the freezer room, T is the average room temperature, and t
is time. Adopting an explicit and forward method for calculating the
thermal inertia M and overall heat gain Q_ Gain is complicated and
requires detailed information of the specific room and stored goods.
On the other hand, an inverse method makes an estimation of these
two parameters possible based on measurements of room temperature and an analysis of the system performance. Fig. 5 summarizes
the steps for the proposed analysis approach from data collection to
the estimation of instantaneous thermal inertia and total heat gain.
Identifying
pieces of
monotonic
variation
Identifying
temperature set
points
Identifying
temperature
swinging
regime areas
Fig. 5. Summarized algorithm of the proposed model.
Calculating
thermal inertia
and heat gain
values
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In the first step, the room temperature is measured. Averaging
and noises in temperature measurements can lead to a nonsmooth temperature diagram. To ensure robust computations, it
is important to first smooth the temperature fluctuations. Many
noise reduction and smoothing algorithms are available in the literature. In this work, weighted averaging techniques similar to the
one used in Smoothed Particle Hydrodynamics (SPH) [21] is used.
The same technique is used to calculate the time derivative of the
average temperature. The smoothing process may create inaccuracies, but is necessary for future steps where calculation of time
derivatives is needed. Fig. 6 shows the smoothed temperature as
well as its calculated time derivative for a sample piece of the data.
A numerical algorithm is developed that sweeps the data to find
the time steps where the temperature derivative approaches zero.
These time steps signify a change in the increasing or decreasing
trend of the smoothed temperature. Once these extrema are found,
the algorithm divides the temperature diagram into ‘‘pieces’’ which
fall between them. Each piece is either monotonically increasing or
decreasing. Fig. 7 shows a few of these data pieces identified by the
algorithm. In order to present the piece-identification process
more clearly, an exponential curve fit is provided for each of the
pieces shown in Fig. 7. The specific period shown in Fig. 7 contains
16 pieces demonstrated by exponential correlations. These 16
pieces are numbered in Fig. 7. The pieces can be categorized into
the following:
Pull-down process
The data pieces where the temperature is decreasing are called
pull-down processes. During pull-down processes, cooling effect is
provided by the refrigeration unit to compensate the instantaneous
heat gains and the thermal inertia of the freezer room. Therefore,
the average room temperature is pulled down.
Heat-gain process
0
-13
-4
-13.5
-6
-8
-14
-10
-12
-14
-14.5
dT / dt (×0.0001 °C/s)
Room Temperature, T (°C)
-2
-16
Room Temperature
Smoothed Temperature
Temperature Derivative (dT / dt)
-15
0
10
20
30
-18
-20
40
50
60
Time, t (Min)
Fig. 6. Smoothing of the raw temperature and its calculated derivative with respect
to time. The diagram covers the data for 1 September 2013 from 00:01 am until
01:00 am (see Fig. 3).
Room Temperature
Piecewise Correlation
DT Q_ Gain ¼ M Dt Heatgain
DT Q_ Cooling þ Q_ Gain ¼ M Dt Pulldown
-10
7
-12
8
-14 1
5
3
2
4
11 13 15
6
9
10
12 14 16
-16
0
50
100
150
ð2Þ
since the refrigeration unit does not provide any cooling effect during the heat-gain period. Meanwhile, during the pull-down processes of the swinging regimes, cooling load is also provided to
the freezer room. Therefore, the heat balance equation for any
pull-down process takes the following form:
-8
Room Temperature, T (°C)
The data pieces in which temperature is increasing are called
heat-gain processes. Note that during heat gain, the refrigeration
unit can be on or off depending on the room temperature status
compared to the set points. During this process, the heat gain surpasses the potential cooling effect provided by the refrigeration
unit, which in turn results in an increase in the average room
temperature.
As previously discussed, the temperature diagram contains
periods of freezer operation during which the temperature swings
between the low and high set points. These oscillations are identified by consecutive pull-down and heat-gain processes. We also
know that the temperature is below the high set point during
any heat-gain process in the swinging regime. Therefore, the cooling system is turned off during such processes in the swinging
periods. This is a key observation that allows the calculation of
thermal inertia under such circumstances.
Assume a heat-gain process and its consecutive pull-down process within a temperature swinging period demonstrated in Fig. 4.
The heat balance in Eq. (1) for the heat-gain process can be simplified to:
200
Time, t (Min)
Fig. 7. Sample piece of temperature data for demonstration of pull-down and heatgain processes. The data is divided into pieces as shown by piecewise exponential
curve fits that are numbered in the figure. Temperature variations on 2 September
2013 from 00:01 am until 04:00 am are shown.
ð3Þ
where Q_ Cooling has a negative value for cooling. Averaging over the
time period of each piece, the time derivative dT/dt of Eq. (1) is
replaced by DT/Dt. Thus, a constant overall heat gain value can be
calculated from Eqs. (2) and (3). DT = T2 T1 and Dt = t2 t1 are
the bulk changes in temperature and time during the process,
respectively. Based on the definition, it is evident that DT/Dt > 0
for heat-gain processes and DT/Dt < 0 for pull-down processes.
Considering that the swinging regime occurs mostly at times
when there is no change in the freezer room constituents (usually
during nights when the door is shut and no goods are loaded or
unloaded), we can assume that the thermal inertia of the system
remains constant during the two processes of Eqs. (2) and (3). Since
the door remains shut, the ventilation load is negligible during
swinging regimes. The variation of temperature difference
between the freezer room and the ambient air is also negligible,
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M.A. Fayazbakhsh et al. / Applied Energy 138 (2015) 496–504
Q_ Cooling
M ¼ DT DDTt Heatgain
Dt Pull—down
ð4Þ
Once the thermal inertia is known for a specific time, it is further used to find the average heat gain using Eq. (2) for heat-gain
processes and Eq. (3) for pull-down processes.
Evaporator Outlet
Evaporator Inlet
-12
Evaporator Temperature (°C)
since both the inside and outside temperatures are almost constant. This results in a relatively constant heat gain due to direct
and ambient heat loads. Therefore, it can be reasonably assumed
that the heat gain is constant between every two consecutive
heat-gain and pull-down processes of a swinging period. Thus,
knowing the cooling effect provided by the refrigeration unit, one
can solve Eqs. (2) and (3) to arrive at the following relationship
for the lumped thermal inertia of the freezer room:
-13
Evaporator Inlet
-14
-15
-16
-17
Evaporator Outlet
-18
-19
4. Results and discussion
0
500
1000
1500
2000
2500
Time, t (s)
Fig. 8. Air temperatures at the inlet and outlet of the evaporator during an instance
of temperature swing between the high and low set points.
The volumetric flow rate of the evaporator fan is measured to be
600 CFM which is equivalent to a mass flow rate of 0.38 kg in the
corresponding temperature range shown in Fig. 8. The sensible
cooling capacity is thus calculated as:
_ p ðT o T i Þ
Q_ Cooling ¼ mc
ð5Þ
_ is the air mass flow rate, cp is the air specific heat, To is the
where m
evaporator outlet air temperature, and Ti is the evaporator inlet air
temperature. Fig. 9 shows the cooling capacity calculated based on
the data shown in Fig. 8 and Eq. (5). As observed in Fig. 9, the cooling capacity is zero during the heat-gain processes. But during the
pull-down processes, the cooling capacity oscillates around a constant value of almost Q_ Cooling ¼ 950 W. The cooling capacity in
Fig. 9 does not show a significant trend or variation during the
swing regime. Accordingly, the excessive calculations and measurements required for the instantaneous cooling load can be avoided
and a constant cooling capacity can be used in relevant applications
of the proposed method without remarkable loss of accuracy. The
same assumption is used in the rest of this study.
200
0
Cooling Power (W)
The analysis approach and the temperature data shown in Fig. 3
are implemented in a computer code for calculation of thermal
inertia and heat gain. The algorithm detects periods of temperature
swings. Once these periods are identified, the code calculates the
room thermal inertia.
In order to use Eq. (4), it is necessary to know the cooling load
Q_ Cooling provided by the evaporator. In general, the provided cooling
load varies by both the evaporator and condenser coil temperatures. Nevertheless, Eq. (4) is only applied during the mentioned
swinging regimes when the evaporating temperature is oscillating
in a narrow range between the set points. As a result, whenever the
value of Q_ Cooling is used in the present method, it merely represents
the average cooling capacity of the system operating between the
high and low set points. Jabardo et al. [22] and Wang et al. [23]
showed that the cooling capacity varies by less than 5% due to
1 °C of evaporating temperature increase. Both studies show the
same behavior for the dependence of the cooling capacity to the
condensing temperature. Thus, although the cooling capacity is a
function of the evaporating and condensing temperatures, its variation can be neglected for small changes in those temperatures.
Based on on-site measurements and the manufacturer information, the approximate cooling capacity provided by the studied
refrigeration unit is Q_ Cooling ¼ 950 W while the freezer temperature is swinging within the high and low set points. Since the condenser is placed in a restaurant indoor space, the condensing
temperature is maintained relatively constant. Furthermore, since
the set points are only 1.3 °C apart, as observed in Fig. 4, the evaporator also experiences a relatively constant temperature during
the temperature swinging regimes. Thus, the freezer performance
is negligibly affected by the variations of evaporator and condenser
temperatures, and the provided cooling capacity can be assumed
constant at Q_ Cooling ¼ 950 W. The negative sign shows the direction of heat transfer from the room to the outside environment.
To verify the assumption of constant cooling load, further measurements are performed on the studied refrigeration cycle. Fig. 8
shows the air temperature at the inlet and outlet of the evaporator
fan as measured by T-type thermocouples (5SRTC-TT-T-30–36,
OMEGAÒ) during the same period of study (28 August 2013–3 September 2013). The thermocouples have an accuracy of ±1 °C and
are installed at the inlet and outlet of the evaporator fan. Fig. 8
shows the measured evaporator temperatures for an arbitrary
instance of the swings demonstrated in Fig. 4. When the cooling
system is turned off, the inlet and outlet evaporator temperatures
are almost equal and a heat-gain process occurs. During the cooldown processes, on the other hand, there is almost a constant
gap between the temperatures which hints a relatively constant
cooling load.
-200
-400
-600
-800
-1000
-1200
0
500
1000
1500
2000
2500
Time, t (s)
Fig. 9. Cooling power provided by the refrigeration system during an instance of
temperature swing between the high and low set points.
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15
700
10
600
500
5
400
0
300
-5
200
-10
Thermal Inertia, M (kJ/°C)
Room Temperature, T (°C)
Room Temperature
Thermal Inertia
100
-15
0
0
1
2
3
4
5
6
7
Time, t (Day)
Fig. 10. Thermal inertia calculation results for the freezer room from 28 August
2013 at 00:01 am until 3 September 2013 at 11:59 pm.
Fig. 10 shows the calculation results for room thermal inertia
during the period under consideration. Since the freezer is regularly used by the restaurant personnel for storing foodstuff, the
contents of the room vary during the daytime. It can be inferred
from the temperature variations of Fig. 10 that during daytime
the freezer room experienced several events of door opening,
which resulted in random temperature values far above the set
points. Therefore, the swinging heat-gain and pull-down processes,
occurring during night times, have been used to estimate the thermal inertia until the next occurrence of temperature swing pattern.
Fig. 11 shows the estimated heat gain values. The heat gains are
calculated for every time step based on the thermal inertia information inferred from Fig. 10 and Eqs. (2) and (3). For the sake of
clarity, only results covering the first 12 h of the refrigeration unit
operation on 2 September 2013 are shown in Fig. 11. As mentioned
in Fig. 3, there are 3 different temperature variation regimes:
swinging regimes, temperature spikes, and heavy duty periods.
Room Temperature
Heat Gain
15
Room Temperature, T (°C)
5
2000
0
-5
0
Heat Gain, Q (W)
4000
10
-10
-15
-2000
0
2
4
6
8
10
12
Time, t (Hour)
Fig. 11. Heat gain variations calculated on 2 September 2013 from 00:01 am until
12:01 pm. See Figs. 3 and 8 for temperature and thermal inertia information during
the entire week of study.
The data shown in Fig. 11 are chosen so they contain the 3 different
patterns in order to show the capability of the present method for
handling all of them.
It is noticeable in Fig. 11 that the defrost events that occur at
hours 1 and 7 impose significant amounts of load on the room.
The instantaneous heat gain values jump to above 4 kW during
defrost events which result in fast increase of temperature in the
freezer room. As previously mentioned, the defrost events are set
to automatically occur every 6 h with no monitoring and sensing
of ice formation on the evaporator coils. Nevertheless, there is a
considerable energy-saving potential in using intelligent defrost
units [24].
According to Fig. 11, during the temperature swings of hour 1 as
well as hours 3–7, the heat gain is estimated to be at almost the
same level as the cooling power provided by the refrigeration
cycle. As a result, the temperature is kept at a relatively constant
level in a swinging manner. During the defrost spikes of hours 2
and 8, there are 2 distinct heat-gain and pull-down processes. In
the heat-gain section of defrost events, there is a large 4 kW heat
load imposed on the freezer from the heater-based defrost system.
On the other hand, during the pull-down section of defrost events,
there is less heat gain imposed on the freezer, since the defrost
heater is off. Furthermore, notice that the heat gain is a direct function of the temperature difference between the freezer air and
ambient air. At the peak of the defrost spike, there is a minimum
difference between the inside and ambient air temperatures. As a
result, the amount of heat gain in the pull-down section of a
defrost event is even less than the steady heat gain of swinging
regimes.
The heavy duty part of Fig. 11, covering hours 8–12, demonstrates random variations in the heat gain level. During this period
of time, several cases occur when the restaurant personnel open
the freezer door and allow large heat transfers from the ambient
air into the freezer through a convective ventilation mechanism.
Nevertheless, whenever the door is closed, the temperature is
decreased and the amount of heat gain is also reduced to the levels
governed by wall heat fluxes.
Table 1 lists the daily-averaged thermal inertia and heat gain
values of the freezer room. The present inverse method allows
the identification of the usage pattern in the freezer room based
on temperature measurements. For instance, as listed in Table 1,
some goods are added to the freezer room on 29 August 2013
resulting in an increase of the overall room thermal inertia by
70 kJ/°C compared to the previous day. Such information that are
inferred from temperature measurements can help retrofit existing
systems in real-time while knowing little else about the freezer.
Daily-averaged heat gain values are also reported in Table 1. It is
apparent that relatively higher lumped heat gains are encountered
during 31 August 2013. Consistent with this observation is the fact
that in Fig. 3, many more door opening occurrences were observed
in the first 4 days of measurements.
A detailed list of foodstuff stored in the freezer room with their
corresponding weight and thermal inertia is prepared based on
Table 1
Calculated daily-averaged thermal inertia and heat gain in the freezer room.
Date
Daily-averaged thermal
inertia (kJ/°C)
Daily-averaged heat
gain (W)
28 August 2013
29 August 2013
30 August 2013
31 August 2013
1 September 2013
2 September 2013
3 September 2013
Average
519
589
606
599
566
573
588
577
1872
2113
2034
2041
1711
1751
1753
1896
503
M.A. Fayazbakhsh et al. / Applied Energy 138 (2015) 496–504
Table 2
Measured mass and thermal inertia values for the freezer contents on 31 August
2013.
a
Product
Mass (kg)
Thermal inertia (kJ/°C)
Chicken cages
Dry ribs
Sliced pepperoni
Sockeye
Shrimp
Halibut
Lobster
Calamari squid
Crab meat chunky
Albacore loin
Fries
Spicy chorizo
Sweet potato fries
Hash browns
Multi grain bread
White bread
Pesto
Corn
Peas
Raspberries
Red tortilla
Green tortilla
Whole wheat tortilla
Flour tortilla
Gluten free pizza shells
Gluten free buns
Brioche buns
Chocolate shaving
Vanilla ice cream
Miscellaneous parts
Total
25.00
25.00
5.00
10.00
10.00
25.00
6.00
25.00
25.00
5.00
15.00
5.00
7.50
15.00
2.10
3.80
3.00
12.00
12.00
5.00
0.02
0.02
0.02
25.00
25.00
25.00
25.00
2.50
11.00
100.00a
455
43.00
38.50
7.75
15.70
17.20
41.75
10.32
42.00
43.00
8.80
21.15
7.75
11.93
21.15
3.47
6.27
5.64
17.04
11.04
8.80
0.04
0.04
0.04
41.25
40.00
40.00
42.50
3.18
18.37
60.00a
628
Estimated value.
to improve the validation. Assuming a combined mass of 100 kg
for the miscellaneous objects including the evaporator coils and
shelf structures, and using an average specific heat of 0.6 kJ/kg °C
for the metallic components, the total miscellaneous thermal inertia is calculated as 60 kJ/°C. This value is added to the total freezer
thermal inertia in Table 2.
The total measured value of the room bulk thermal inertia is
628 kJ/°C, while the calculated average value of thermal inertia
for the whole week of study is 577 kJ/°C. Hence, there is a discrepancy of 8% between the calculated and measured thermal inertia.
There is generally no direct method for measuring the heat gain
in an air conditioning or refrigeration application. Typically, a heat
balance of the room alongside appropriate correlations is used to
estimate the amount of heat gain prior to the system design. The
results of these thermal analyses are often estimations of the heat
gain and provide an approximate value that has acceptable accuracy for the specific application at hand. Following the same
approach, geometrical, material, and thermal properties of the
freezer room are measured and summarized in Table 3. Convective
heat transfer coefficients are estimated using correlations from
ASHRAE [7] for turbulent natural convection on vertical and horizontal flat plates. The walls are equipped with old polyurethane
insulation which is degraded due to several years of operation,
hence providing a poor thermal resistance. The degradation mechanisms can cause the polyurethane thermal resistance to decrease
to half of its original value [25]. Based on the methodology of the
heat balance method [7], the heat transfer across the closed walls
of the room is estimated. The overall heat gain of the room air is
thus equal to the summation of these wall heat fluxes:
Q_ Gain ¼
on-site measurements. Existing tabulated values in ASHRAE Handbook of Fundamentals [7] are used for calculation of thermal inertia. The collected data is used to validate the proposed inverse
modelling approach. Table 2 shows the list of foodstuff stored in
the freezer room during 31 August 2013. The sum of all measured
thermal inertia values is considered as the total thermal inertia of
the freezer room. Due to the demand-based addition and withdrawal of foods, the freezer contents have slightly changed during
the week of study, but they are kept in roughly constant amounts
to ensure steady fulfilment of kitchen orders. Thus, although these
are the values of one sample day, they are also deemed to represent an average of the freezer contents for the whole week.
Both the mass and the specific heat of the miscellaneous objects
such as the evaporator, lights, shelves, and boxes are considerably
smaller than those of the foodstuff. For instance, the specific heat
of carbon steel, aluminum, and copper are 0.49 kJ/kg °C, 0.91 kJ/
kg °C, and 0.39 kJ/kg °C, respectively, and these values are considerably smaller than the thermal inertia of white bread and corn
which are 1.65 kJ/kg °C and 1.42 kJ/kg °C, respectively. As a result,
the miscellaneous thermal inertia can be neglected in many applications without significant loss of accuracy. Nonetheless, an
estimation of the miscellaneous thermal inertia is calculated and
added to the measured thermal inertia of the foodstuff in order
X
AðT Amb T Þ
1=h
o þ b=k þ 1=hi
Walls
ð6Þ
where A is the wall surface area, TAmb is the ambient temperature, T
is the room average temperature, b is the wall thickness, k is the
average wall thermal conductivity, and ho and hi are the outside
and inside convection heat transfer coefficients, respectively.
As observed in Fig. 4, the high and low set points are 13.8 °C
and 15.1 °C, respectively. In order to find an approximate temperature difference between the ambient and room temperature during the swinging regimes, we assume the room temperature to be
at the average of the set points, i.e., T = 14.5 °C. The restaurant
temperature was measured at several locations near the freezer
room and the average value of TAmb = 22.0 °C was obtained. Using
the properties collected in Table 3, Eq. (6) yields to a total ambient
heat gain value of Q_ Gain ¼ 1002 W. During the swinging regions of
Fig. 11, the calculated heat gain varies approximately between
1050 W and 1250 W, with an average value of Q_ Gain ¼ 1150 W.
Thus, although the heat gain and room temperature vary with
time, a comparison of the average calculated value with the heat
gain acquired from the heat balance analysis of the freezer room
shows an error of less than 15%.
A future step in using the calculated results such as shown in
Figs. 8 and 9 can be to utilize them in a dynamic control algorithm
that modifies the supplied cooling load by the refrigeration unit
based on the real-time thermal inertia and heat gain information.
Table 3
Geometrical, material, and thermal properties of the freezer room.
Location
Left
Back
Right
Front
Roof
Floor
Surface area, A (m2)
Outside convective coefficient, ho (W/m2 °C)
Inside convective coefficient, hi (W/m2 °C)
Insulation thickness, b (cm)
Wall thermal conductivity, k (W/m °C)
6.7
2.8
2.8
3
0.06
5.7
2.8
2.8
3
0.06
6.7
2.8
2.8
3
0.06
5.7
2.8
2.8
3
0.06
6.2
0.6
0.6
3
0.06
6.2
3.1
3.1
3
0.06
504
M.A. Fayazbakhsh et al. / Applied Energy 138 (2015) 496–504
Since the algorithm provides a real-time estimation of the instantaneous heat gain, a refrigeration unit equipped with a variablespeed compressor can be intelligently controlled to provide the
instantaneous required cooling load to the freezer room. Such control algorithms might be able to reduce the annual energy consumption of air conditioning and refrigeration systems.
5. Conclusions
A new inverse method was proposed to estimate the real-time
thermal inertia and heat gain in air conditioning and refrigeration
systems based on on-site temperature measurements. The collected temperature data were smoothed and fed to a mathematical
algorithm that detects periods of temperature swing between the
set points. The pace and pattern of temperature variations during
the swing regimes were utilized to calculate the thermal inertia
and overall heat gain in the freezer room. Little information on
the geometry, material, and usage pattern of the system were used
which made the proposed algorithm ideal for inverse analysis and
retrofit of existing refrigeration and air conditioning systems.
The algorithm was validated using experimental data collected
from a walk-in freezer room of a restaurant in Surrey, British
Columbia, Canada during a week of its regular operation. It was
shown that several duty patterns could be recognized only by analyzing the temperature data over time. The inverse approach
enabled us to interpret detailed information on the usage pattern
of the freezer and to calculate the thermal parameters of the system. The method can be implemented in control systems of refrigeration units to reduce the overall energy consumption of
stationary and mobile HVAC-R units.
Acknowledgements
This work was supported by Automotive Partnership Canada
(APC), grant APCPJ 401826-10. The authors would like to thank
the kind support of the Central City Brew Pub, 13450 102nd avenue, Surrey, British Columbia, Canada.
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